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In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressibility as power series, and satisfying the Cauchy integral formula. In real analysis, it is usually more natural to consider differentiable, smooth, or harmonic functions ...
Let be a function in the Lebesgue space ([,]).We say that in ([,]) is a weak derivative of if ′ = ()for all infinitely differentiable functions with () = =.. Generalizing to dimensions, if and are in the space () of locally integrable functions for some open set, and if is a multi-index, we say that is the -weak derivative of if
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when
When more specific types of differentiation are necessary, such as in multivariate calculus or tensor analysis, other notations are common. For a function f of a single independent variable x , we can express the derivative using subscripts of the independent variable:
The following notation will be used throughout this article: is a fixed positive integer and is a fixed non-empty open subset of Euclidean space. = {,,, …} denotes the natural numbers.
In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions.